In this case, given a Borel function h, we can define an operator h(T) by specifying its behavior on the basis:

In general, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator

acting on L2 of some measure space. The domain of T consists of those functions for which the above expression is in L2. In this case, we can define analogously

For many technical purposes, the preceding formulation is good enough. However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of T as a multiplication operator. This we do in the next section.

The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator T, the existence of a Borel functional calculus can be shown in an elementary way as follows:

is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity defines f(T) for a continuous function f on the spectrum of T. The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus.

Alternatively, the continuous calculus can be obtained via the Gelfand transform, in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation, T can be a normal operator.

Given an operator T, the range of the continuous functional calculus h → h(T) is the (abelian) C*-algebra C(T) generated by T. The Borel functional calculus has a larger range, that is the closure of C(T) in the weak operator topology, a (still abelian) von Neumann algebra.

We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.

Theorem. Let T be a self-adjoint operator on H, h a real-valued Borel function on R. There is a unique operator S such that

The operator S of the previous theorem is denoted h(T).

More generally, a Borel functional calculus also exists for (bounded) normal operators.

Let T be a self-adjoint operator. If E is a Borel subset of R, and 1E is the indicator function of E, then 1E(T) is a self-adjoint projection on H. Then mapping

is a projection-valued measure called the resolution of the identity for the self adjoint operator T. The measure of R with respect to Ω is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral . Sometimes the term "resolution of the identity" is also used to describe this representation of the identity operator as a spectral integral.

In the case of a discrete measure (in particular, when H is finite-dimensional), can be written as

in the Dirac notation, where each is a normalized eigenvector of T. The set is an orthonormal basis of H.

In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as

and speak of a "continuous basis", or "continuum of basis states", Mathematically, unless rigorous justifications are given, this expression is purely formal.